beyondbinaryfandomcom-20200215-history
Unsolved Problems
(coming soon) |YouTube:/Edward Frenkel/Mysteries of Math and the Langlands Program (4 episodes broadcast on the Japanese TV channel NHK)> "A series of 4 lectures by Edward Frenkel on the Mysteries of Mathematics filmed at MSRI, Berkeley and broadcast on the Japanese TV channel NHK in the Fall of 2015 in the "Luminous Classroom" series. The lectures went from elementary topics such as Pythagoras theorem, prime numbers and symmetries to Fermat's last theorem and the general Langlands conjectures, and to the recent work connecting the Langlands Program to Quantum Physics. Even though the Intro is in Japanese, the lecture itself is in English." Mathematics Hilbert Problems https://en.wikipedia.org/wiki/Hilbert%27s_problems https://en.wikipedia.org/wiki/Hilbert%27s_sixth_problem "Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent. It occurs on the widely cited list of Hilbert's problems in mathematics that he presented in the year 1900." "David Hilbert himself devoted much of his research to the sixth problem;3 in particular, he worked in those fields of physics that arose after he stated the problem. In the 1910s, celestial mechanics evolved into general relativity. Hilbert and his assistant Emmy Noether corresponded extensively with Albert Einstein on the formulation of the theory.4 In the 1920s, mechanics of microscopic systems evolved into quantum mechanics. Hilbert, with the assistance of John von Neumann, L. Nordheim, and E. P. Wigner, worked on the axiomatic basis of quantum mechanics (see Hilbert space).5 At the same time, but independently, Dirac formulated quantum mechanics in a way that is close to an axiomatic system, as did Hermann Weyl with the assistance of Erwin Schrödinger. In the 1930s, probability theory was put on an axiomatic basis by Andrey Kolmogorov, using measure theory. Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, also modern quantum field theory can be considered close to an axiomatic description." Millennium Problems https://en.wikipedia.org/wiki/Millennium_Prize_Problems http://www.claymath.org/millennium-problems "Yang–Mills and Mass Gap Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known. Riemann Hypothesis The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2. P vs NP Problem If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution. Navier–Stokes Equation This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding. Hodge Conjecture The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown. Poincaré Conjecture In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture. Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries. Birch and Swinnerton-Dyer Conjecture Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three." Riemann Hypothesis (see also Category:Analytic Number Theory#Riemann Hypothesis) :"In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part ½. Many consider it to be the most important unsolved problem in pure mathematics (Bombieri 2000). It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named. :The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems." Physics Quantum Gravity (see Quantum Gravity) https://en.wikipedia.org/wiki/Hilbert%27s_sixth_problem#Status "At the present time, there are two foundational theories in physics: the Standard Model of particle physics and general relativity. Many parts of these theories have been put on an axiomatic basis. However, physics as a whole has not, and in fact the Standard Model is not even logically consistent with general relativity, indicating the need for a still unknown theory of quantum gravity." Category:Mathematics Category:Physics